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Black String solutions in Rainbow Gravity

In this paper we study black string solutions under the consideration of rainbow gravity. We have analytically obtained the solution for four-dimensional black strings in terms of the functions $f(E/E_p)$ and $g(E/E_p)$ that sets the energy scale where the rainbow gravity becomes relevant. We have also obtained the Hawking temperature for the black string, from which we could see that the rainbow functions play the role of increasing or decreasing the Hawking temperature for a given horizon radius depending on the choice of such rainbow functions. We have computed the entropy, specific heat and free energy for the black string. The entropy and specific heat exhibit a rainbow dependence, while the free energy is not modified by the rainbow functions. Finally we have studied the effects of the rainbow gravity in the orbits of massive and massless particles around a black string. We could verify that neither massive nor massless particles exhibit stable orbits around a black string in the scenario of rainbow gravity, for any configuration of rainbow functions.